We’ve seen her in concert halls, on TV — singing “Hallelujah” at the 2010 Olympics — even on the cover of Vanity Fair, getting a close shave from Cindy Crawford. But until this week, no one’s seen k.d. lang on Broadway.

“I never thought in a million years I’d be doing this,” the 52-year-old singer tells The Post. In a peacoat and jeans, hair cropped short, she looks like a big, soft boy. She seems rattled — she is, after all, stepping into the Duke Ellington musical “After Midnight,” taking over for Fantasia. Here’s what the singer’s saying about music, love — and fellow Canadian Justin Bieber.

Has Broadway beckoned before?

Early on, in my [1987 debut album] “Angel with a Lariat” days, they did approach me to do “Annie Get Your Gun.” But I never had Broadway ambitions. It was out of my comfort zone. I decided I’d give [“After Midnight”] a try because the show’s legitimately based in music. But I don’t know what’s gonna happen.

Did Tony Bennett, your favorite duet partner, give you any tips?

He has! “Make sure you sing ‘Stormy Weather’ — that’s one of the greatest songs in the history of American traditional music!” He’s a huge fan of Duke Ellington. They were friends, I’m pretty sure.

So what’s hardest these days: being lesbian, vegetarian or Canadian?

First of all, they’re inseparable! But it depends. If you’re in trouble, then a Canadian; if you’re in a steakhouse, a vegetarian; and if you’re in a strip bar . . . probably a Buddhist.

Any advice for your countryman, Justin Bieber?

I like it that Justin Bieber’s Canadian when he’s in a s - - t ton of trouble; when he’s being fabulous, he’s American! I’ve got lots of advice for him, but it’s nothing I’d say through the media. I can’t imagine all the influences he’s under now. It’s a lot for anyone to navigate, let alone a 19-year-old.

How did you keep your head during your “Constant Craving” heyday ?

Actually, I don’t think I did keep my head. I didn’t do drugs or drink when I was working so hard, but I think pride and ego and entitlement come naturally from being afforded things you shouldn’t [be] or aren’t usually afforded. That’s the real poison.

You recently moved to Portland, Ore. Why? And will we see you on the TV show “Portlandia”?

As a matter of fact, yes. I’ll be on “Portlandia” this month! I’d been touring for 30 years, and every time I went to Portland, I felt a weird comfort. After living in LA for 21 years, I realized it was time to move on.

Did you once say you were against gay marriage?

I don’t think I was ever against it, but I didn’t find it a necessity . . . I was wrong. I’m all for gay marriage, though it’s not something I’m personally interested in.

You recently ended a longtime relationship. Still single?

I’m no longer single! It’s brand new and I don’t know how much I want to publicly announce it, but I’m seeing someone and it’s really fun.